Abstract
Makhnev and Samoilenko have found parameters of strongly regular graphs with no more than 1000 vertices, which may be neighborhoods of vertices in antipodal distance-regular graph of diameter 3 and with \(\lambda=\mu\). They proposed the program of investigation vertex-symmetric antipodal distance-regular graphs of diameter 3 with \(\lambda=\mu\), in which neighborhoods of vertices are strongly regular. In this paper we consider neighborhoods of vertices with parameters \((25,8,3,2)\).
Highlights
We consider undirected graphs without loops and multiple edges
There is provided a research program of the study of vertex-symmetric antipodal distance-regular graphs of diameter 3 with λ = μ, in which neighborhoods of vertices are strongly regular with parameters from Proposition 1
Let Γ be a distance-regular graph with intersection array {25, 16, 1; 1, 8, 25}, G = Aut(Γ), g is an element of prime order p in G and Ω = Fix(g) contains exactly s vertices in t antipodal classes
Summary
We consider undirected graphs without loops and multiple edges. Given a vertex a in a graph. There is provided a research program of the study of vertex-symmetric antipodal distance-regular graphs of diameter 3 with λ = μ, in which neighborhoods of vertices are strongly regular with parameters from Proposition 1. In [6] it is proved that distance-regular locally 5 × 5-grid of diameter more 2 is either isomorphic to the Johnson’s graph J(10, 5) or has an intersection array {25, 16, 1; 1, 8, 25}. Let Γ be a distance-regular graph with intersection array {25, 16, 1; 1, 8, 25}, G = Aut(Γ), g is an element of prime order p in G and Ω = Fix(g) contains exactly s vertices in t antipodal classes. Let Γ be a distance-regular graph with intersection array {25, 16, 1; 1, 8, 25} and a group G = Aut(Γ) acts transitively on the set of vertices of Γ.
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